The Conformal Anomaly and the Neutral Currents Sector of the Standard Model ∗ Claudio Corianò , Luigi Delle Rose, Antonio Quintavalle, Mirko Serino

Physics Letters B 700 (2011) 29–38

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Physics Letters B

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The conformal anomaly and the neutral currents sector of the Standard Model ∗ Claudio Corianò , Luigi Delle Rose, Antonio Quintavalle, Mirko Serino

Departimento di Fisica, Università del Salento and INFN-Lecce, Via Arnesano 73100, Lecce, Italy article info abstract

Article history: We elaborate on the structure of the graviton–gauge–gauge vertex in the electroweak theory, obtained Received 10 January 2011 by the insertion of the complete energy–momentum tensor (T ) on 2-point functions of neutral gauge Received in revised form 16 April 2011 currents (VV ). The vertex deﬁnes the leading contribution to the effective action which accounts for Accepted 25 April 2011 the conformal anomaly and related interaction between the Standard Model and gravity. The energy– Available online 4 May 2011 momentum tensor is derived from the curved spacetime Lagrangian in the linearized gravitational limit, Editor: A. Ringwald and with the inclusion of the term of improvement of a conformally coupled Higgs sector. As in the Keywords: previous cases of QED and QCD, we ﬁnd that the conformal anomaly induces an effective massless scalar Conformal anomalies interaction between gravity and the neutral currents in each gauge invariant component of the vertex. Electroweak corrections with external This is described by the exchange of an anomaly pole. We show that for a spontaneously broken theory gravity the anomaly can be entirely attributed to the poles only for a conformally coupled Higgs scalar. In the Effective actions exchange of a graviton, the trace part of the corresponding interaction can be interpreted as due to an effective dilaton, using a local version of the effective action. We discuss the implications of the anomalous Ward identity for the TVV correlator for the structure of the gauge/gauge/effective dilaton vertex in the effective action. The analogy between these effective interactions and those related to the radion in theories with large extra dimensions is pointed out. © 2011 Elsevier B.V. All rights reserved.

1. Introduction son. We will be stating these identities omitting any proof, since the details of the derivations are quite involved. The explicit computation of these radiative corrections (i.e. of Gravity couples to the Standard Model, in the weak gravita- the anomalous action) finds two direct applications. The first has tional field limit, via its energy–momentum tensor (EMT) T μν . to do with the analysis of anomaly mediation as a possible mech- This interaction is responsible for the generation of the radiative anism to describe the interaction between a hypothetical hidden breaking of scale invariance [1–3], which is mediated, at leading sector and the fields of the Standard Model, as shown in Fig. 1(a). order in the gauge coupling (O (g2)), by a triangle diagram: the One of the results of our analysis, in this context, is that anomaly TVV vertex (see [4–7]), where V , V denote two gauge bosons. mediation is described by the exchange of anomaly poles in each The computation of the vertex is rather involved, due to the very gauge invariant sector of the perturbative expansion, as shown in lengthy expression of the EMT in the electroweak theory, but also Fig. 1(b). This feature, already present in the QED and QCD cases, not so obvious, due to the need to extract the correct external con- as we will comment below, is indeed confirmed by the direct straints which are necessary for its consistent definition. computation in the entire electroweak theory. One of the main im- The constraints take the form of 3 Ward identities derived by plications of our analysis, in fact, is that this picture remains valid the conservation of the EMT and of (at least) 3 Slavnov–Taylor even in the presence of mass corrections due to symmetry break- identities (STTs) on the gauge currents. All of them need to be ing, for a graviton of large virtuality and a conformally coupled checked in perturbation theory in a given regularization scheme, Higgs sector. We will comment on this point in a separate section in order to secure the consistency of the result. In the case under (Section 5) and in our summary before the conclusions. exam they correspond to the TAA, TAZ and TZZ vertices, where A second area where these corrections may turn useful is in A is the photon and Z the neutral massive electroweak gauge bo- the case of an electroweak theory formulated in scenarios with large extra dimensions (LED), with a reduced scale for gravity. In this case the virtual exchanges of gravitons provide sizeable cor- Corresponding author. * rections to electroweak processes – beyond tree level – useful for E-mail addresses: [email protected] (C. Corianò), [email protected] (L. Delle Rose), [email protected] LHC studies of these models, as illustrated in Fig. 2 in the case (A. Quintavalle), [email protected] (M. Serino). of the qq¯ annihilation channel. In these extensions a graviscalar

Fig. 1. Gravitational interaction of the Standard Model ﬁelds with a hidden sector (H.S.), at leading order in the gravitational constant (a). The interaction in perturbation theory responsible for the trace anomaly is illustrated in (b) via the exchange of an anomaly pole.

Fig. 2. Typical leading order (O (κ2)) contributions to the production of two gauge bosons with gravitational mediation. Not included are the initial state (Standard Model) corrections on the qq¯/graviton vertex and the loops of gauge bosons and Higgs mediating the decay of the graviton. The latter contribute to the conformal anomaly.

(radion) φ degree of freedom is induced by the compactification, with the symmetric rank-2 tensor hμν(x) accounting for the metric μ which is expected to couple to the anomaly (φTμ )aswellasto fluctuations. the scaling-violating terms, as we are going to clarify, by an extra We denote with Tμν the complete (quantum) EMT of the elec- prescription. This prescription is based on the replacement of the troweak sector of the Standard Model. This includes the contribu- classical trace of the matter EMT by its quantum average. A rig- tions of all the physical fields and of the Goldstones and ghosts in orous discussion of the fundamental anomalous Ward identity for the broken electroweak phase. Its expression is uniquely given by the TVV correlator will clarify some subtle issues involved in this the coupling of the Standard Model Lagrangian to gravity, modulo prescription. We will show, in parallel, that the anomalous effec- the terms of improvements, which depend on the choice of the tive action induces in the 1-graviton exchange channel a similar coupling of the Higgs doublets. As we have mentioned, we have interaction. This interaction can be thought as being mediated by chosen a conformally coupled Higgs field. Our computation is per- an effective massless dilaton, coupled to the trace anomaly equa- formed in the Rξ gauge. The expression of the EMT is symmetric Min tion (and to its mass corrections). and conserved. It is therefore given by a minimal contribution Tμν I (without improvement) and the improvement EMT, Tμν ,with 2. Definitions and Ward and Slavnov–Taylor identities = Min + I Tμν Tμν Tμν, (4) We start with few definitions, focusing our discussions only on the case of the graviton/photon/photon (TAA) and graviton/Z/Z where the minimal tensor is decomposed into (TZZ)vertices. Min = f .s. + ferm. + Higgs + Yukawa + g.fix. + ghost We recall that the fundamental action describing the gravity Tμν Tμν Tμν Tμν Tμν Tμν Tμν . (5) and the Standard Model is defined by the three contributions The various contributions refer, respectively, to the gauge kinetic terms (field strength, f .s.), the fermions, the Higgs, Yukawa, gauge S = S + S + S G SM I fixing contributions (g.fix.) and the contributions coming from the 1 √ √ =− d4x −gR + d4x −gL ghost sector. As we have already mentioned, in order to fix the 2 SM κ structure of the correlator one needs to derive and implement √ the necessary Ward and STTs. Their derivation is quite lengthy as 1 4 † + d x −gRH H, (1) is their implementation in perturbation theory, given the sizeable 6 number of diagrams involved in the expansion and the very long 2 where κ = 16π G N ,withG N being the four-dimensional Newton’s expression of the vertex extracted from the EMT. constant and H is the Higgs doublet. We have denoted with SG the We obtain: contribution from gravity (Einstein–Hilbert term) while SSM is the 1) A Ward identity related to the conservation of the EMT in μν Standard Model (SM) quantum action, extended to curved space- the flat spacetime limit (i.e. ∂μT = 0), which takes the form time. S I denotes the term of improvement for the scalars, which κ are coupled to the metric via its scalar curvature R. The factor − μ i ∂ Tμν(x)Vα(x1)V β (x2) amp 1/6 should be recognized as giving a conformally coupled Lorentz 2 κ scalar, the SU(2) Higgs doublet. The EMT in our conventions is de- =− − (4) − −1VV ∂νδ (x1 x)Pαβ (x2, x) fined as 2 − − ∂ δ(4)(x − x)P 1VV (x , x) 2 δ[SSM + S I ] ν 2 αβ 1 Tμν(x) = √ , (2) −g(x) δgμν(x) + μ (4) − −1VV ∂ ηανδ (x1 x)Pβμ (x2, x) and around a flat spacetime limit + (4) − −1VV ηβνδ (x2 x)Pαμ (x1, x) , gμν(x) = ημν + κhμν(x), (3) where we have introduced the off-diagonal 2-point function C. Corianò et al. / Physics Letters B 700 (2011) 29–38 31

Fig. 3. Amplitudes with the triangle topology for the two correlators TAA and TZZ.

−1VV = | | Pαβ (x1, x2) 0 TVα(x1)V β (x2) 0 amp, (6) 4) A STI for the TAZ vertex where amp denotes amputated external gauge lines. Notice that 1 A Z Tμν(z)F (x)F (y) the gravitational field, in this computation, is just an external field ξ and the 1PI conditions apply only to the external gauge lines. This i σ (4) A =− −ημν∂ δ (z − y) Zσ (z)F (x) point emerges from a closer investigation of the defining STTs of ξ z the correlator. This Ward identity applies to any gauge boson in + ∂ zδ(4)(z − y) Z (z)F A (x) the neutral sector. ν μ 2) A STI for the TAAvertex. Specifically, introducing the photon + z (4) − F A ∂μδ (z y) Zν(z) (x) . (11) gauge-fixing function We illustrate the overall structure of the results for the TAA A σ F = ∂ Aσ , (7) and TZZ vertices, focusing on the essential parts, and in particular on those form factors which contribute to the trace part, since they we obtain the relation are simpler. The complete result is indeed quite involved and some details can be found in [8]. 1 A A Tμν(z)F (x)F (y) ξ 3. Results for the TAAcase i ρ (4) A =− ημν∂ δ (z − x) Aρ(x)F (y) ξ x (AA)μναβ In the TAA case, we introduce the notation Γ (p, q) − ρ (4) − F A to denote the one-loop amputated vertex function with a graviton ημν∂z δ (z y) Aρ(z) (x) and two on-shell photons. x (4) A − ∂ δ (z − x) Aν(x)F (y) In momentum space we indicate with k the momentum of the μ z (4) A incoming graviton and with p and q the momenta of the two pho- − ∂ δ (z − y) A (z)F (x) + (μ ↔ ν) , (8) μ ν tons. In general, the Γ (VV )μναβ (p, q) correlator is defined as with ξ denoting the gauge-fixing parameter. (2π)4δ(4)(k − p − q)Γ VV (p, q) 3) A STI for the TZZ correlator. Introducing the gauge-fixing μναβ function of the Z gauge boson κ 4 4 4 −ikz+ipx+iqy =−i d zd xd y Tμν(z)Vα(x)V (y) e . 2 β amp F Z = ∂σ Z − ξ M φ, (9) σ Z (12) where φ is the Goldstone of the Z, this takes the form In the 2-photon case (AA) is decomposed in the form 1 Z Z (AA)μναβ (AA)μναβ (AA)μναβ Tμν(z)F (x)F (y) Γ (p, q) = Γ (p, q) + Γ (p, q) ξ F B (AA)μναβ + Γ (p, q), (13) i 2 (4) (4) I =− −iξ ημνδ (x − y)δ (x − z) ξ as a sum of a fermion sector (F) (Fig. 3(a), Fig. 4(a)), a gauge bo- ρ + (4) x − z Z x F Z y son sector (B) (Fig. 3(b)–(g), Fig. 4(b)–(g), Fig. 5, Fig. 6) and a term ημν∂x δ ( ) ρ( ) ( ) μναβ of improvement denoted as Γ . The contributions to the (F) x (4) Z I − ∂ δ (x − z) Zν(x)F (y) Min μ and (B) sectors are obtained by the insertion of T . The contri- − x (4) − F Z bution from the term of improvement is given by diagrams of the ∂ν δ (x z) Zμ(x) (y) same form of those in Fig. 3(c), 3(e) and Fig. 5(b), but now with + z (4) − F Z ∂μ δ (z y) Zν(z) (x) the graviton–scalar–scalar vertices determined only by the energy– μν z (4) Z momentum tensor T . + ∂ δ (z − y) Zμ(z)F (x) I ν The tensor basis on which we expand the vertex is given by − ρ (4) − F Z ημν∂z δ (z y) Zρ(z) (x) . (10) four independent tensor structures 32 C. Corianò et al. / Physics Letters B 700 (2011) 29–38

Fig. 4. Amplitudes with t-bubble topology for the correlators TAA and TZZ.

Fig. 5. Amplitudes with s-bubble topology for the correlators TAA and TZZ. Fig. 6. Amplitudes with the tadpole topology for the correlators TAA and TZZ.

μναβ φ (p, q) = sημν − kμkν uαβ (p, q), (14) In the TAA vertex, the contribution to the trace anomaly in the 1 μναβ fermion sector comes from Φ1F which is given by φ (p, q) =−2uαβ (p, q) sημν + 2 pμ pν + qμqν 2 μ ν μ ν 2 − 4 p q + q p , (15) Φ1F s, 0, 0,m f 2 μναβ μ ν ν μ αβ s αν βμ αμ βν 4m φ (p, q) = p q + p q η + η η + η η κ α 2 2 f 3 2 =−i Q − + 2 3π s f 3 s s f − ημν ηαβ − qα pβ 2 4m2 2 2 2 2 f − 2m C0 s, 0, 0,m ,m ,m 1 − . (21) − ηβν pμ + ηβμ pν qα − ηανqμ + ηαμqν pβ , f f f f s

μναβ = μν − μ ν αβ = 2 φ4 (p, q) sη k k η (16) Here we have introduced the QED coupling α e /(4π) and the function where uαβ (p, q) has been defined as 1 a + 1 C 2 2 2 = 2 3 αβ αβ α β 0 s, 0, 0,m ,m ,m log (22) u (p, q) ≡ (p · q)η − q p , (17) 2s a3 − 1 μναβ μναβ among which only φ and φ show manifestly a trace, the obtained from the scalar triangle integral, with 1 4 remaining ones being traceless. A complete computation gives for the various gauge invariant subsectors 4m2 a3 = 1 − . (23) s 3 (AA)μναβ = 2 μναβ ΓF (p, q) ΦiF s, 0, 0,m f φi (p, q), (18) Thesumistakenoverallthefermions(f ) of the Standard Model. i=1 As one can immediately realize, this form factor is characterized 3 by the presence of an anomaly pole (AA)μναβ = 2 μναβ Γ (p, q) ΦiB s, 0, 0, MW φ (p, q), (19) α B i Φ F ≡ iκ Q 2 (24) i=1 1,pole 9π s f f (AA)μναβ = 2 μναβ ΓI (p, q) Φ1I s, 0, 0, MW φ1 (p, q) which is responsible for the generation of the anomaly in the + 2 μναβ Φ4I s, 0, 0, MW φ4 (p, q). (20) massless limit. To appreciate the significance of this “pole contribution” one needs special care, since a computation of the residue The first three arguments of the form factors stand for the three (at s = 0) shows that this is indeed zero in the presence of mass independent kinematical invariants k2 = (p + q)2 = s, p2 = q2 = 0 corrections. However, this leading 1 s behaviour in the trace part while the remaining ones denote the particle masses circulating in / of the amplitude, as we are going to show, is clearly identifiable the loop. We use the on-shell renormalization scheme. in an (asymptotic) expansion (s m2 ), and is corrected by extra As already shown in the QED and QCD cases [5–7],inanun- f 2 2 broken gauge theory the entire contribution to the trace anomaly m f /s terms, where m f denotes generically any fermion of the SM. comes from the first tensor structure φ1. In other words, this component is extracted in the UV limit of the C. Corianò et al. / Physics Letters B 700 (2011) 29–38 33 amplitude even in the massive case and is a clear manifestation of generated in the broken phase of the theory. This separation of the the anomaly. radiative from the explicit contributions to the breaking of confor- The other gauge-invariant sector of the TAA vertex is the one mal invariance, due to the tree-level mass terms, is in agreement mediated by the exchange of bosons, Goldstones and ghosts in the with the obvious fact that in the UV limit, masses can be dropped. loop.WewilldenotewithMW , M Z and M H the masses of the W At the same time the (radiative) breaking of the conformal sym- and Z gauge bosons and the Higgs mass respectively. In this sector metryremains,withnomuchsurprise. the form factor contributing to the trace is Preliminarily, we recall that in the MS scheme the β functions of the Standard Model are given by 2 Φ1B s, 0, 0, M W 3 2 g1 20 1 κ α 5 2M β1 = ng + , =−i − W 16π 2 9 6 2 π s 6 s g3 4 22 1 2M2 β = 2 n − + , 2 2 2 2 W 2 2 g + 2M C0 s, 0, 0, M , M , M 1 − , (25) 16π 3 3 6 W W W W s 3 g3 4 which multiplies the tensor structure φ , responsible for the gen- β3 = −11 + ng , (33) 1 16 2 3 eration of the anomalous trace. π In this case the anomaly pole is easily isolated from (25) in the for the hypercharge, weak and strong interactions respectively, form and ng is the number of generations. The expression of the β κ α 5 function of the electromagnetic coupling, βe ,isgiveninthesame Φ1B,pole ≡−i . (26) scheme by 2 π s 6 3 The term of improvement is responsible for the generation of two 2 2 e 32 βe = c β1 + s β2 = ng − 7 . (34) form factors, both of them contributing to the trace. They are given w w 16π 2 9 by At this point, the residue of the anomaly pole which appears 2 in the form factors Φ1,F , Φ1,B and Φ1,I is uniquely determined by Φ1I s, 0, 0, MW the beta function of the electromagnetic coupling constant. Indeed κ α 2 2 2 2 =−i 1 + 2M C0 s, 0, 0, M , M , M , (27) we have 2 3π s W W W W

2 = κ α 2 2 2 2 Φ = Φ F + Φ B + Φ I Φ4I s, 0, 0, MW i MW C0 s, 0, 0, MW , MW , MW , (28) 1,pole 1,pole 1,pole 1,pole 2 6π κ α 2 5 κ βe the ﬁrst of them being characterized by an anomaly pole =−i − Q 2 + + 1 = i , (35) 2 3π s 3 f 2 3s e κ α f Φ1I,pole =−i . (29) 2 3π s 2 = 8 where we have used the fact that f Q f 3 ng . Our considerations on the UV behaviour of the a) radiative plus the b) explicit mass corrections to the anomalous amplitude are ob- 4. Results for the TZZcase viously based on an exact computation of the correlator. In the asymptotic limit (s →∞), the expansions of the three Moving to the vertex with two massive Z gauge bosons, one form factors contributing to the trace part can be organized in discovers a similar pattern. Also in this case, as before, we intro- terms of the 1/s “pole component” plus mass corrections, which duce the notation Γ (ZZ)μναβ (p, q) to describe the corresponding are given by correlation function. We have several contributions appearing in the global expression of the correlator: 2 Φ1,F s, 0, 0,m f (ZZ)μναβ = (ZZ)μναβ + (ZZ)μναβ 2 2 Γ (p, q) ΓF (p, q) ΓW (p, q) κ α 2 m f m f − 2 − + + 2 − 2 (ZZ)μναβ (ZZ)μναβ i Q f 4 π log + + 2 3π s 3 s s ΓZ,H (p, q) ΓI (p, q). (36) f (ZZ)μναβ 2 Γ (p, q), for on-shell Z bosons, can be separated into three m f Min − 2iπ log , (30) contributions obtained using the insertion of T (sectors F , W s and Z/H) and a fourth one coming from the term of improvement. 2 In this case the gravitational interaction is mediated by T I . Φ1,B s, 0, 0, MW The labelling of the ﬁrst three is inherited from the types of 2 2 κ α 5 M M particles (and corresponding masses) that circulate in the loops. −i − W 2 + π 2 − log2 W 2 π s 6 s s Beside the fermion sector (F) with diagrams depicted in Figs. 3(a) and 4(a), the other contributions involve a W gauge boson (sector M2 − 2iπ log W , (31) (W )), with diagrams Fig. 3(b)–(g), Fig. 4(b)–(g), Fig. 5 and Fig. 6, s and the mixed Z/Higgs bosons sector (Z, H) with contributions 2 shown in Figs. 7, 8, 9 and 10. There is also a diagram propor- Φ1,I s, 0, 0, MW tional to a Higgs tadpole (Fig. 10(a)) which vanishes in the on-shell 2 2 2 κ α M M renormalization scheme. Finally there is a contribution from the −i 1 − W π − i log W . (32) 2 3π s s s term of improvement (I). This is given by the diagrams depicted in Fig. 3(c), (d), 5(b), together with those of Figs. 7(b), (c), (d) and 2 2 2 2 The energy suppressed terms (m f /s , MW /s ) take the typical Fig. 9. In this case, however, the graviton–scalar–scalar vertices is 2 2 μν form M /s ,withM denoting, generically, any explicit mass term generated by T I . 34 C. Corianò et al. / Physics Letters B 700 (2011) 29–38

Fig. 7. Amplitudes with the triangle topology for the correlator TZZ.

As we have already mentioned, we take the two Z gauge bosons on the external lines on-shell, and an insertion of Tμν at a nonzero momentum transfer k. The four contributions can be ex- panded on a tensor basis given by 9 tensors, and corresponding form factors Φi as 9 (ZZ)μναβ = (F ) 2 2 2 μναβ ΓF (p, q) Φi s, M Z , M Z ,m f ti (p, q), (37) Fig. 8. Amplitudes with the t-bubble topology for the correlator TZZ. i=1 9 (ZZ)μναβ = (W ) 2 2 2 μναβ ΓW (p, q) Φi s, M Z , M Z , MW ti (p, q), (38) i=1 (ZZ)μναβ ΓZ,H (p, q) 9 = (Z,H) 2 2 2 2 μναβ Φi s, M Z , M Z , M Z , M H ti (p, q), (39) i=1 (ZZ)μναβ Fig. 9. Amplitudes with the s-bubble topology for the correlator TZZ. Γ (p, q) I (I) μναβ = Φ s, M2 , M2 , M2 , M2 , M2 t (p, q) 1 Z Z W Z H 1 + (I) 2 2 2 2 2 μναβ Φ2 s, M Z , M Z , MW , M Z , M H t2 (p, q), (40) where the first three arguments of the Φi ’s are the virtualities of 2 = 2 = 2 = 2 the external lines k s, p q M Z , while the last two give the masses in the internal lines. 7 of the 9 tensor structures are traceless, while the only two responsible for the breaking of scale invariance are Fig. 10. Amplitudes with tadpole topology for the correlator TZZ. s μναβ = μν − μ ν − 2 αβ − α β t (p, q) sg k k M Z g q p , 1 2 (Z,H) 7iακ Φ ≡ . (44) 1 pole 2 2 μναβ = μν − μ ν αβ 144π scw sw t2 (p, q) sg k k g . (41) The term of improvement contributes to two tensor structures but The four form factors responsible for generating a pole term are only one of the two form factors from this sector has a pole term. those accompanying the tensor structure t , while the form fac- 1 We have, in this case tors Φ2, corresponding to the tensor structure t2,shownopole. 2 2 (I) κ α The latter give contributions which are suppressed as M /s . Φ =−i 1 − 2s2 c2 , (45) 1 pole 2 2 w w Therefore, as for the TAA vertex, the trace parts show a distinctive 2 6π sw cw s 1/s contribution plus corrections of O (M2/s2),aswehavespeci- (I) ∼ 2 2 fied above. Being the complete result of this vertex quite lengthy, while Φ4 MW /s asymptotically. we omit details and just focus our attention on the pole terms ex- The coefficient of the anomalous pole contribution is fixed by tracted from each sector. These are summarized by rather simple the beta functions of the theory. In this case it is proportional to expressions. We obtain a linear combination of the beta functions of the couplings g1 and g of hypercharge and SU(2).Indeedwehave 2 (F ) iακ f 2 f 2 Φ ≡ C + C (42) (F ) (W ) (Z,H) (I) 1pole 2 2 a v Φ = Φ + Φ + Φ + Φ 36πc s s 1,pole 1,pole 1,pole 1,pole 1,pole w w f β β = κ 2 1 + 2 2 for the fermion sector of the vertex, where sw and cw are short i sw cw . (46) 3s g1 g2 notations for sin θW and cos θW . Similarly, in the other sectors we have 5. The coupling of the radion/dilaton beyond tree level and the 4 − 2 + effective dilaton (W ) κ α (60s 148s 81) Φ ≡−i w w (43) 1 pole 2 s2 c2 π s 72 w w One of the most significant applications of the results of for diagrams involving W ’s, while the diagrams with W and Z the previous sections concerns the study of the coupling of the gauge bosons give radion/dilaton to the fields of the Standard Model in theories C. Corianò et al. / Physics Letters B 700 (2011) 29–38 35 with LED. We will use the term radion (φ) to denote the funda- on the external lines, due to the SU(2) and SU(3) field strengths mental scalar introduced in the usual compactifications of theories (F2, F3). with LED, and reserve the name of “effective dilaton” (ϕ)forthe In the context of theories with extra dimensions, the correlator μ scalar interaction dynamically induced by the anomaly. As we are obtained by the insertion of the trace Tμ plays a key role in de- going to show, the radion has interactions with matter which are scribing the radiative corrections to the tree-level coupling of φ to quite similar to those allowed to the effective dilaton, although the matter. We present here the explicit form of this vertex when the latter shows up in a different channel (the 1-graviton exchange radion couples to on-shell external photons. It is defined as channel). We proceed first with a rigorous discussion of the inter- (2π)4δ(4)(k − p − q)D AA(p, q) action of the LED radion and then illustrate the analogies between αβ the two states to clarify these points. κ 4 4 4 μ −ikz+ipx+iqy We recall that in models with LED, with matter on the brane =−i d zd xd y Tμ (z)Aα(x)Aβ (y) e (51) 2 amp and gravity in the bulk, the compactification of the extra dimensions gives rise in the four-dimensional effective field theory to (with an amputated correlation function) and can be decomposed towers of Kaluza–Klein gravitons and dilatons. For definiteness in the form we consider a theory compactified on a torus and consider the D(AA)αβ zero modes of the 4D graviton field and of the dilaton φ gen- (p, q) (AA)αβ (AA)αβ (AA)αβ erated by this procedure. These two fields will couple, via their = D (p, q) + D (p, q) + D (p, q), (52) lowest Kaluza–Klein modes, to the EMT with the interaction La- F B I grangian [9] where (AA)αβ 2 D (p, q) L =−κ 4 μν + μ = F int d x hμν T ωφTμ , ω , (47) 2 2 3(δ + 2) 4 4m =− κ α 2 2 + f − i Q f m f 2 1 where δ is the number of extra dimensions. To understand the 2 π s s f main features of the dilaton interaction at 1-loop level we proceed as follows. 2 2 2 αβ × C0 s, 0, 0,m ,m ,m u (p, q), (53) We first recall that the structure of the anomaly equation in the f f f presence of a classical trace in a certain theory takes the form (AA)αβ D (p, q) B μν μ η Tμν(z) = A(z) + Tμ (z) , (48) M2 =− κ α 2 − W C 2 2 2 i 6MW 1 2 0 s, 0, 0, MW , MW , MW where we have taken the quantum average of each term. A is 2 π s the operator describing the anomalous behavior of the fields under 2 μ MW αβ scale transformations while the Tμ operator is the non-anomalous − 6 − 1 u (p, q), (54) contribution to the trace of the EMT. This second term vanishes s in the conformal limit (i.e. before electroweak symmetry breaking) D(AA)αβ I (p, q) using the equations of motion of the fields. In an exact gauge the- κ α 2 2 2 2 αβ ory the expected structure of the anomaly is given by the relation =−i 1 + 2M C0 s, 0, 0, M , M , M u (p, q) 2 π W W W W β κ α s A = i αβ i + i M2 C s, 0, 0, M2 , M2 , M2 αβ (55) Fi Fαβ , (49) W 0 W W W η 2gi 2 π 2 i correspond to the contributions coming from the insertion on αβ where Fi and gi are the field strengths and the gauge couplings the photon 2-point function of the trace of the EMT, as specified of the gauge fields in the unbroken phase, corresponding to the in (51). These correspond to fermion (F ) and boson (B) loops, to- Standard Model gauge group SU(3)C × SU(2)L × U (1)Y .Forathe- gether with terms of improvement (I). A description of these terms ory in a broken phase, and in the photon case, the anomaly A is can be found in [10]. again proportional to βe . By taking two functional derivatives of Note that these expressions are ultraviolet finite and do not the trace identity (48) with respect to the sources Jα and Jβ of need any renormalization counterterm. One can also observe the ± κ α the gauge fields Vα and V β , we obtain the anomalous identities presence of two scaleless terms in Eqs. (54) and (55) (the i 2 π s on the correlation functions analyzed in this work terms), which do not depend on any mass parameter but only on 1/s. These are not part of the anomaly – since the D’s correspond μν η Tμν(z)Vα(x)V β (y) to explicit breaking of the conformal symmetry – and seem to in- 2 validate our argument about the pole origin of the entire anomaly δ A(z) μ = + T (z)V (x)V (y) . (50) for a spontaneously broken theory. However these extra scaleless α β μ α β δ J (x)δ J (y) contributions, as one can easily check, cancel in (52) if the Higgs The first term on the right-hand side of the equation above defines scalar is conformally coupled, since they appear with the opposite the residue of the anomaly pole that we have already discussed sign. D AA and isolated in the previous sections. The second term, instead, We can summarize this analysis by saying that αβ is zero is the correlation function obtained by inserting the trace of the for a conformal theory (e.g. QED with massless fermions) and it EMT on the two point functions Vα(x)V β (y) (with the inclu- is expected to be proportional to any mass parameter of the the- sion of terms of gauge fixings and ghosts). This would be the ory otherwise. For instance it is nonzero for QCD and QED when only contribution describing the explicit breaking of the confor- the quarks are massive. Indeed one can explicitly check, for ex- Dgg mal symmetry – in the absence of an anomalous breaking induced ample, that in the QCD case the corresponding amplitude αβ , by the radiative corrections. It is also evident from the structure of coming from the insertion of the trace of the EMT on the gluon Eq. (49) that the complete anomalous effective action takes con- 2-point function, even if not zero, does not contribute any scale- tributions from vertex functions with two and three gauge bosons less term on the right-hand side of the anomaly equation (Eq. (48) 36 C. Corianò et al. / Physics Letters B 700 (2011) 29–38 or (50)). The same is true in the electroweak theory only if the 1 μ ρ ∗α ∗β − T μV ραβ (p, q) (p) (q), (56) Higgs doublet is conformally coupled to gravity. In our case this is n − 2 guaranteed – by construction – due to the specific choice of the co- 2 (0) κ ∗ ∗ efficient in front of the term of improvement. If the improvement M =− 2 μ 2 ρ α β dil ω T μ P k V ραβ (p, q) (p) (q) had not been included in the EMT, then this would have implied 4 2 2 that extra scaleless contributions had to combine with the pole κ ω μ ρ ∗α ∗β =− T μV ραβ (p, q) (p) (q), (57) term in Eq. (26) to saturate the anomaly. This is equivalent to say- 4 k2 ing that the pole term in the correlator, in this specific case, would where the (p), (q) are the polarization vectors of the two final not be entirely responsible for the generation of the anomaly. In- state photons and −i κ V ρσαβ (p, q) is the graviton–two photons deed, for a conformally coupled Higgs only the sum of Eqs. (26) 2 vertex (MV = 0) and (29) encloses the entire contribution to the anomaly, which thus can be entirely attributed to the pole part. κ ρσαβ κ 2 μναβ μναβ It is important to observe that if the definition of the coupling −i V (p, q) =−i k1 · k2 + M C + D (k1,k2) 2 2 V of the dilaton φ to the trace of the EMT is four-dimensional, then there is no coupling of the same state to the anomaly. This is 1 + Eμναβ (k ,k ) (58) indeed the content of Eq. (52), which does not include any anoma- ξ 1 2 lous term of the form φ FF generated by the classical Lagrangian geometrically reduced on the brane. For this reason, the coupling with of the dilaton to the anomaly is obtained only if we make one ex- Cμνρσ = gμρ gνσ + gμσ gνρ − gμν gρσ , tra assumption. μ ρ For instance, in our formulation we need to replace the φTμ D (k ,k ) = g k k − gμσ kνk + g k k μν μνρσ 1 2 μν 1σ 2ρ 1 2 μρ 1σ 2ν vertex appearing in (47) with the vertex φg Tμν at the onset, and then use Eq. (48). Notice that in this expression the EMT does − gρσk1μk2ν + (μ ↔ ν) , not need to be renormalized. In fact, one can show explicitly that Eμνρσ (k1,k2) = gμν(k1ρk1σ + k2ρk2σ + k1ρk2σ ) the renormalization does not affect the trace of the same tensor, being the counterterm vertex in TJJ proportional to a traceless − gνσk1μk1ρ + gνρk2μk2σ + (μ ↔ ν) . (59) form factor [4,7]. For this reason the operation of trace on T μν μν μνρσ 2 2 (i.e. g Tμν) can be computed by the insertion of the bare EMT P (k ) and P(k ) are the (massless) graviton and dilaton prop- in 2-point functions. agators in the de Donder gauge which are given by, in the frame- In other approaches the same coupling requires a redefinition work of dimensional regularization (n = 4 − ), of the trace of the EMT from 4 to D dimensions. In this second i 2 case the renormalization of the trace operator is essential in or- iPμνρσ k2 = ημρ ηνσ + ημσ ηνρ − ημνησρ , 2 der to generate the coupling of the dilaton to the complete scale k n − 2 violations (anomaly plus explicit terms) present in the anomaly i μ iP k2 = . (60) equation. This is obtained by the replacement in (47) of φTμ (in 2 μ μ k 4 dimensions) with φTr μD , where Tr μ is the trace of the renor- Now we consider the one-loop corrections to these expressions malizedEMT,computedinD dimensions [11,12]. and introduce the notation We are now going to briefly discuss and compare the structure of the effective scalar interactions obtained from the trace (AA) κ (AA) Γ (p, q) =−i Γ¯ (p, q), anomaly pole against those coming from the exchange of a fun- μναβ 2 μναβ damental radion introduced by a generic extra dimensional model. (AA) κ (AA) D (p, q) =−i D¯ (p, q), (61) An effective degree of freedom in the form of a dilaton (ϕ FF) αβ 2 αβ interacting both with the anomaly and with the (explicit) scale violating terms is induced by the effective action generated by the in order to factorize the gravitational coupling constant. We obtain anomaly loop. This effective interaction can be carefully identified 2 1 M(1) =−κ μν ¯ (AA) in the 1-graviton exchange channel not only for a massless theory grav T Γ (p, q) 2 k2 μναβ [4] but also in the presence of explicit scale non-invariant terms. One can investigate the salient features of these interactions by a 1 μ ρσ ¯ (AA) ∗α ∗β − T μη Γ (p, q) (p) (q) direct computation. n − 2 ρσαβ 2 κ 1 (AA) 5.1. Amplitudes for graviton/radion exchange in the production of two =− T μνΓ¯ (p, q) 2 k2 μναβ gauge bosons 1 − μ D¯ (AA) + A ∗α ∗β T μ αβ (p, q) αβ (p, q) (p) (q), For example, let’s consider the production of two photons by a n − 2 gravitational source characterized by a certain EMT Tμν . The tree- (62) level amplitudes with the exchange of the first modes of the KK κ2 ω2 towers, namely a massless graviton and a massless dilaton can be M(1) =− μ D¯ (AA) + A ∗α ∗β dil T μ αβ (p, q) αβ (p, q) (p) (q). formally written as 4 k2 (63) κ2 A M(0) =− μνρσ 2 ∗α ∗β The αβ (p, q) term is the anomaly contribution generated by the grav Tμν P k Vρσαβ (p, q) (p) (q) 4 pole terms in the Γ (AA)(p, q) vertex and it is given by 2 κ 1 β =− T μν V (p, q) e 2 μναβ Aαβ ≡−2 uαβ (p, q). (64) 2 k e C. Corianò et al. / Physics Letters B 700 (2011) 29–38 37

Notice that in (63) we have retained both the coupling of the similar to that of the chiral anomaly [14]. The pole is found only dilaton φ to the explicit (non-conformal) and anomalous terms in the massless case. generated by the Ward identity of the trace anomaly. A similar Obviously, a similar interpretation of the origin of this singular- scalar interaction appears in the graviton channel (proportional to ity, which is present over the entire light cone (s ∼ 0) of the TVV μ Tμ ), as one can easily infer from the right-hand side of Eq. (62). 3-point function, should also hold in the QCD case (in the mass- Both interactions are, indeed, of dilaton type, being proportional to less fermion limit) and should extend also to the gluon loop. In D(AA) + A fact, in QCD, beside the contribution of the fermion sector, a sepa- the complete ( αβ αβ ) trace of the anomaly loop. We will briefly comment on the origin of this effective dilaton interaction. rate gauge invariant contribution comes from the gluon sector [7]. For this purpose we recall [4,7] that in the case of massless It is also important to remark that the direct computation of QED the effective interaction induced by the trace anomaly takes [5,7], here applied to the entire neutral sector of the Standard Model, is not based on a dispersive analysis and gives the exact the form expression of the effective action at 1-loop. For this reason it al- 4 4 (1) −1 μν S ∼ d xd yR (x, y)Fμν(y)F (y) (65) lows to establish the presence of these contributions on a more general ground, both in the massless and in the massive cases, and (1) where R denotes the linearized scalar curvature and Fμν is the is in agreement with the dispersive analysis (in the massless case). abelian field strength. A similar result holds for QCD. As shown in [4] this expression coincides with the long-known anomaly- 6.1. Summary induced action derived by Riegert [13], which was derived for a generic gravitational field, after an expansion of its expression We have indeed seen, in combination with a previous study for around the flat spacetime limit. Notice that in terms of auxiliary QCD [7], that an explicit computation of the exact 1-loop effec- degrees of freedom (i.e. two scalar fields (ϕ,ψ )) which render the tive action (at leading order in the combined gravitational (κ) and action (65) local [4], extra couplings of the form ϕ FF are automat- gauge coupling expansion (g)) shows two fundamental features: ically induced by the 1/ term. This interaction is indeed present 1) In a massless gauge theory the breaking of conformal in- in the equivalent Lagrangian variance is characterized by a typical 1/ behaviour. Notice that this does not exclude the possible appearance of other nonlocal S g, A; ϕ,ψ anom terms in the same effective action, such as those proportional to √ log() (see the discussion in [15]). These additional terms, at least 4 R c αβ = d x −g −ψ ϕ − ψ + Fαβ F ϕ , (66) in the case of the chiral anomaly, are generated by the insertion 3 2 of the triangle diagram into a graph of higher perturbative order (c =−β(e)/(2e)) where ϕ and ψ are the auxiliary scalar fields. [16]. Therefore, in the chiral case, they are not part of the trian- This action does not account for any correction to the trace gle diagram (i.e. of the lowest order contribution to the anomaly). anomaly equation due to the appearance of mass terms. The pres- The computations in QED and QCD of the trace anomaly are in (AA) ence of an explicit breaking of scale invariance due to the D line with this result and are in agreement with Riegert’s anomaly- term, however, can be handled by a modification of the ϕ FF in- induced action [13] in these two theories. This result of ours ap- teraction present in (66), i.e. the anomaly term. In the presence pears to be also in agreement with the observations in [15] to of an explicit breaking of scale invariance, the effective dilation ϕ which we refer for further details. couples to the neutral currents of the final state just like the fun- 2) In a gauge theory in a spontaneously broken phase, such as damental dilaton φ, which is the content of the Eq. (62).Onecan the electroweak theory, the radiative and the explicit breaking of explicitly check the cancellation of possible “double pole” contribu- conformal invariance are separately identifiable in the ultraviolet tions in the s-channel. These could be induced by the (single) pole limit. This result holds even if conformal invariance is broken by of the graviton propagator together with the anomaly pole com- the Higgs vev already at tree-level. ¯ (AA) ing from the triangle loop (present in Γμναβ ) (see Eq. (62)). This Therefore, the two (distinct) kinematical domains characterized cancellation holds under the condition that the source EMT T μν by the dominance of the anomaly (via its massless poles) are de- is conserved, as expected. Indeed this is an additional check of the scribed, in our notations, by a single invariant, s, which denotes significance of this effective component of the 1-graviton exchange the virtuality of the external graviton. In particular, point 1), as we amplitude generated in the presence of a trace anomaly vertex. have already mentioned, is related to the s ∼ 0behaviourofthe anomalous contributions of the TVV vertex (for a massless the- 6. Discussion ory) and point 2) to the s →∞limit of the same correlator (in the massive case). There are some comments which are in order concerning the Hence, in the massive case (point 2), the 1/s contribution ap- result of this analysis, which complete those obtained in the QED pears only after an asymptotic expansion at large energy of the and QCD cases [4,5,7]. From all these investigations it seems clear anomaly vertex, and should be interpreted as its dominant asymp- that anomaly mediation can be described, in a perturbative frame- totic component in the trace part. As such, this component is not work, as due to the exchange of effective massless scalar degrees part of the amplitude in the infrared (i.e. it is not present at s = 0). of freedom between gravity and the gauge sector. The physical in- In fact, the computation of the residue of the correlator at s = 0 terpretation of these singularities is probably easier to grasp by shows that this indeed vanishes when masses are present. In the a dispersive analysis, at least in the massless case, as discussed fermion case, for instance, this result is due to cancellations be- in [4] for QED. In the QED case, in fact, this component is gen- tween the pole and the second and third terms of Eq. (21). erated (diagrammatically) by a virtual graviton decaying into two There is no doubt, however, that the 1/s term, present in the on-shell collinear (correlated) fermions, which later decay into two expansion of the anomalous diagram at large s, is a manifesta- on-shell photons. This interpretation follows from the fact that the tion of the same “anomaly pole” encountered in the infrared in the spectral density of the fermion loop diagrams (ρ(s) ∼ δ(s)), which massless case, since its contribution is asymptotically corrected by indeed generates a pole, is obtained by cutting the graviton → γγ mass effects (M2/s2) which become negligible in the UV limit. It amplitude in the s = (p + q)2 channel, thus setting two intermedi- seems obvious that we should recover the behaviour typical of a ate fermion lines in the triangle diagram on-shell. The approach is masslesstheoryaswemovetohighenergy,andthepoletermof 38 C. Corianò et al. / Physics Letters B 700 (2011) 29–38 the anomaly, as the expansions (32) suggest. We have seen that of the quantum trace of the energy–momentum tensor for these effective dilaton interactions are automatically part of the effective theories. We have discussed two different (but equivalent) ways to action which parallel those introduced within models with large obtain this interaction using extra dimensional models. We have extra dimensions. also shown that the appearance of an effective dilaton – cou- Thus, it could be of interest, for instance, to explore the role pled both to the anomaly and to extra scale-dependent terms – that such contributions could play in the analysis of perturbations is a generic feature of the effective action which accounts for the in the early universe, for instance in the context of inflation driven trace anomaly. This effective interaction can be identified in the by a vector field [17], where such interactions appear to be neces- 1-graviton exchange channel when we couple Einstein gravity in 4 sary. In this case this vertex would be a direct consequence of the dimensions to the Standard Model. We have illustrated the analo- conformal anomaly, without any need to resort to more complex gies between the interaction of the radion and of the effective scenarios for its generation. We have also rigorously shown that dilaton using some examples. a pole term completely accounts for the anomaly, in the Standard Model, only if the Higgs scalar is conformally coupled. In the non- Acknowledgements conformally coupled case, extra scaleless contributions appear in the anomalous Ward identity for the TVV . For a conformally cou- We thank E. Mottola, R. Armillis, E. Dimastrogiovanni, N. Irges pled scalar indeed there is a cancellation between scaleless contri- and M. Karciauskas for discussions. butions coming from the explicit breaking of the conformal symmetry and those generated by the terms of improvement (Eqs. (54) References and (55)). Obviously, this picture is typical only of theories with a spontaneous breaking of the gauge symmetry. For unbroken gauge [1] M. Duff, Nucl. Phys. 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